# Hamiltonian for guiding center motion: symplectic structure approach

## TL;DR

This paper introduces a canonical Hamiltonian framework for guiding center motion in plasma physics, simplifying the analysis of charged particle dynamics in magnetic fields by decoupling different motion types.

## Contribution

It presents an alternative canonical Hamiltonian approach that decouples motion types, improving analysis of plasma systems compared to traditional non-canonical methods.

## Key findings

- Hamiltonian decouples gyrorotation, field-aligned motion, and drifts.
- Allows straightforward inclusion of adiabatic invariants.
- Facilitates analysis of time-dependent electromagnetic fields.

## Abstract

The guiding center approximation represents a very powerful tool for analyzing and modeling a charged particle motion in strong magnetic fields. This approximation is based on conservation of the adiabatic invariant, magnetic moment. Hamiltonian equations for the guiding centre motion are traditionally intoduced using a non-canonical symplectic structure. Such approach requires application of non-canonical Hamiltonian perturbation theory for calculations of the magnetic moment corrections. In this study we present an alternative approach with canonical Hamiltonian equations for guiding centre motion in time-dependent electromagnetic fields. We show that the derived Hamiltonian decouples three types of motion (gyrorotation, field-aligned motion, and across-field drifts), and each type is described by a pair of conjugate variables. This form of Hamiltonian and symplectic structure allows simple introduction of adiabatic invariants and can be useful for analysis of various plasma systems.